I don't think SQL Search is at fault here. I think you've stumbled across Microsoft's sp_rename bug whereby it doesn't update the syscomments entry with the new definition. I'm fairly sure that this was fixed in SSMS 2008. I thought this was fixed in a later edition of SSMS 2005, although I may be wrong there. Can you confirm which version you're using?

I think the conclusion is that the best approach is to avoid the rename feature. Red Gate's SQL Refactor has a 'Smart Rename' feature that does this properly (as well as renaming all references throughout the schema), although this is a commercial tool.